Rooney became the Three Lions' all-time leading scorer when he hammered home a late penalty against Switzerland last month, and the United skipper is now ahead of Red Devils legend Sir Bobby Charlton on 50 goals for his country.
Rooney was in appreciative mood after the game against the Swiss and thanked everyone for their support en route to the milestone, before tipping the likes of Kane and Manchester City winger Raheem Sterling for greatness on the international stage.
Kane burst onto the scene last season and plundered 31 goals for Spurs to earn a place in the national side under Roy Hodgson, and he netted on his debut after less than two minutes against Lithuania back in March.
The goals are yet to flow as freely this term for the Spurs star, but with three strikes already for England, the 22-year-old is planning to eventually usurp Rooney as the Three Lions' leading marksman.
"That was a special night, just be part of history [when Rooney broke the record]," Kane told ITV Sport. "Just be part of the night, but the fact that Rooney mentioned me in his speech after the game [was great]. Hopefully I might one day go and chase the record down.
"We'll see what he ends up on, he's still got a few more years to play, so I'll try and chase him down slowly but surely.
"I seem to be doing OK, I seem to be scoring every time I come away with England. I play in a great side that create a lot of chances and I've been able to take them so far so hopefully I can continue to do that for England.
England's place at Euro 2016 was secured with a perfect 10 as Hodgson's side romped to victory in Group E, and the Three Lions put the icing on their qualifiying campaign over the last few days as they beat Estonia 2-0 on Friday at Wembley before securing a 3-0 victory over Lithuania on Monday.
An impressive 18 months for England has seen them go without defeat since they were beaten by Uruguay in the World Cup back in June 2014, but Hodgson's side will face a stiff test of their capabilities when they face France and Spain in upcoming friendlies ahead of the European Championships next summer.
Kane returns to domestic action on Saturday when Spurs host Liverpool, who roll into White Hart Lane with Jurgen Klopp as their manager and having hammered in eight in their last two appearances at the home of Mauricio Pochettino's men.